Tuesday, March 24, 2009

Endogenous if-then-else

> I am trying to define a nonlinear function where the variable in the
> objective is dependent on a parameter (x) which varies across
> i=1,2,...,N cases and two variables, a cut-off value (cutoff) and a
> coefficient (coeff) which transforms x above the value of cutoff. The
> logic, in pseudocode is:
>
> if x[i] > cutoff {
>     x2[i] = x[i] * coeff
>     }
> else {
>     x2[i] = x[i]
>     }
>
> The objective function would be something like:
>
> minimize Objective: sum {i in N} [complicated function goes here] * x2[i] ;
>
> How, if at all, can this be implemented in AMPL?
>

You could try:
e{i in I}: x2[i] = if (x[i]>cutoff) then x[i]*coeff else x[i];
However this is a discontinuous function so beware that many NLP solvers assume smooth functions and may get into trouble near the jump. A smooth approximation would be better in that case (there are several possible functional forms for such a smooth approximation; one would need to invest some time to try a few of them; an example is shown here). If the model was otherwise linear, one could look into a piecewise linear formulation (AMPL has very good support for this; this approach is less interesting if the model is otherwise nonlinear as it would become an MINLP).